--- interfacial/interfacial.tex	2011/05/09 19:08:08	3725
+++ interfacial/interfacial.tex	2011/06/29 18:14:23	3729
@@ -22,7 +22,7 @@
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@@ -71,7 +71,7 @@ leads to higher interfacial thermal transfer efficienc
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Introduction}
-
+[BACKGROUND FOR INTERFACIAL THERMAL CONDUCTANCE PROBLEM]
 Interfacial thermal conductance is extensively studied both
 experimentally and computationally, and systems with interfaces
 present are generally heterogeneous. Although interfaces are commonly
@@ -97,6 +97,7 @@ There have been many algorithms for computing thermal 
 
 \section{Methodology}
 \subsection{Algorithm}
+[BACKGROUND FOR MD METHODS]
 There have been many algorithms for computing thermal conductivity
 using molecular dynamics simulations. However, interfacial conductance
 is at least an order of magnitude smaller. This would make the
@@ -131,88 +132,212 @@ external thermostat. 
 algorithm conserves momenta and energy and does not depend on an
 external thermostat. 
 
-(wondering how much detail of algorithm should be put here...)
+\subsection{Defining Interfacial Thermal Conductivity $G$}
+For interfaces with a relatively low interfacial conductance, the bulk
+regions on either side of an interface rapidly come to a state in
+which the two phases have relatively homogeneous (but distinct)
+temperatures. The interfacial thermal conductivity $G$ can therefore
+be approximated as:
+\begin{equation}
+G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle -
+    \langle T_\mathrm{cold}\rangle \right)}
+\label{lowG}
+\end{equation}
+where ${E_{total}}$ is the imposed non-physical kinetic energy
+transfer and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle
+  T_\mathrm{cold}\rangle}$ are the average observed temperature of the
+two separated phases.
 
-\subsection{Force Field Parameters}
-Our simulation systems consists of metal gold lattice slab solvated by
-organic solvents. In order to study the role of capping agents in
-interfacial thermal conductance, butanethiol is chosen to cover gold
-surfaces in comparison to no capping agent present.
+When the interfacial conductance is {\it not} small, two ways can be
+used to define $G$.
 
-The Au-Au interactions in metal lattice slab is described by the
-quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC
-potentials include zero-point quantum corrections and are
-reparametrized for accurate surface energies compared to the
-Sutton-Chen potentials\cite{Chen90}. 
-
-Straight chain {\it n}-hexane and aromatic toluene are respectively
-used as solvents. For hexane, both United-Atom\cite{TraPPE-UA.alkanes}
-and All-Atom\cite{OPLSAA} force fields are used for comparison; for
-toluene, United-Atom\cite{TraPPE-UA.alkylbenzenes} force fields are
-used with rigid body constraints applied. (maybe needs more details
-about rigid body)
-
-Buatnethiol molecules are used as capping agent for some of our
-simulations. United-Atom\cite{TraPPE-UA.thiols} and All-Atom models
-are respectively used corresponding to the force field type of
-solvent.
-
-To describe the interactions between metal Au and non-metal capping
-agent and solvent, we refer to Vlugt\cite{vlugt:cpc2007154} and derive
-other interactions which are not parametrized in their work. (can add
-hautman and klein's paper here and more discussion; need to put
-aromatic-metal interaction approximation here)
-
-[TABULATED FORCE FIELD PARAMETERS NEEDED]
-
-\section{Computational Details}
-\subsection{System Geometry}
-Our simulation systems consists of a lattice Au slab with the (111)
-surface perpendicular to the $z$-axis, and a solvent layer between the
-periodic Au slabs along the $z$-axis. To set up the interfacial
-system, the Au slab is first equilibrated without solvent under room
-pressure and a desired temperature. After the metal slab is
-equilibrated, United-Atom or All-Atom butanethiols are replicated on
-the Au surface, each occupying the (??) among three Au atoms, and is
-equilibrated under NVT ensemble. According to (CITATION), the maximal
-thiol capacity on Au surface is $1/3$ of the total number of surface
-Au atoms. 
-
-\cite{packmol}
-
-\subsection{Simulation Parameters}
-
-When the interfacial conductance is {\it not} small, there are two
-ways to define $G$. If we assume the temperature is discretely
-different on two sides of the interface, $G$ can be calculated with
-the thermal flux applied $J$ and the temperature difference measured
-$\Delta T$ as:
+One way is to assume the temperature is discretely different on two
+sides of the interface, $G$ can be calculated with the thermal flux
+applied $J$ and the maximum temperature difference measured along the
+thermal gradient max($\Delta T$), which occurs at the interface, as:
 \begin{equation}
 G=\frac{J}{\Delta T}
 \label{discreteG}
 \end{equation}
-We can as well assume a continuous temperature profile along the
-thermal gradient axis $z$ and define $G$ as the change of bulk thermal
-conductivity $\lambda$ at a defined interfacial point:
+
+The other approach is to assume a continuous temperature profile along
+the thermal gradient axis (e.g. $z$) and define $G$ at the point where
+the magnitude of thermal conductivity $\lambda$ change reach its
+maximum, given that $\lambda$ is well-defined throughout the space:
 \begin{equation}
 G^\prime = \Big|\frac{\partial\lambda}{\partial z}\Big|
          = \Big|\frac{\partial}{\partial z}\left(-J_z\Big/
            \left(\frac{\partial T}{\partial z}\right)\right)\Big|
-         = J_z\Big|\frac{\partial^2 T}{\partial z^2}\Big|
+         = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big|
          \Big/\left(\frac{\partial T}{\partial z}\right)^2
 \label{derivativeG}
 \end{equation}
+
 With the temperature profile obtained from simulations, one is able to
 approximate the first and second derivatives of $T$ with finite
 difference method and thus calculate $G^\prime$.
 
 In what follows, both definitions are used for calculation and comparison.
 
+[IMPOSE G DEFINITION INTO OUR SYSTEMS]
+To facilitate the use of the above definitions in calculating $G$ and
+$G^\prime$, we have a metal slab with its (111) surfaces perpendicular
+to the $z$-axis of our simulation cells. With or withour capping
+agents on the surfaces, the metal slab is solvated with organic
+solvents, as illustrated in Figure \ref{demoPic}.
+
+\begin{figure}
+\includegraphics[width=\linewidth]{demoPic}
+\caption{A sample showing how a metal slab has its (111) surface
+  covered by capping agent molecules and solvated by hexane.}
+\label{demoPic}
+\end{figure}
+
+With a simulation cell setup following the above manner, one is able
+to equilibrate the system and impose an unphysical thermal flux
+between the liquid and the metal phase with the NIVS algorithm. Under
+a stablized thermal gradient induced by periodically applying the
+unphysical flux, one is able to obtain a temperature profile and the
+physical thermal flux corresponding to it, which equals to the
+unphysical flux applied by NIVS. These data enables the evaluation of
+the interfacial thermal conductance of a surface. Figure \ref{gradT}
+is an example how those stablized thermal gradient can be used to
+obtain the 1st and 2nd derivatives of the temperature profile.
+
+\begin{figure}
+\includegraphics[width=\linewidth]{gradT}
+\caption{The 1st and 2nd derivatives of temperature profile can be
+  obtained with finite difference approximation.}
+\label{gradT}
+\end{figure}
+
+\section{Computational Details}
+\subsection{System Geometry}
+In our simulations, Au is used to construct a metal slab with bare
+(111) surface perpendicular to the $z$-axis. Different slab thickness
+(layer numbers of Au) are simulated. This metal slab is first
+equilibrated under normal pressure (1 atm) and a desired
+temperature. After equilibration, butanethiol is used as the capping
+agent molecule to cover the bare Au (111) surfaces evenly. The sulfur
+atoms in the butanethiol molecules would occupy the three-fold sites
+of the surfaces, and the maximal butanethiol capacity on Au surface is
+$1/3$ of the total number of surface Au atoms[CITATION]. A series of
+different coverage surfaces is investigated in order to study the
+relation between coverage and conductance.
+
+[COVERAGE DISCRIPTION] However, since the interactions between surface
+Au and butanethiol is non-bonded, the capping agent molecules are
+allowed to migrate to an empty neighbor three-fold site during a
+simulation. Therefore, the initial configuration would not severely
+affect the sampling of a variety of configurations of the same
+coverage, and the final conductance measurement would be an average
+effect of these configurations explored in the simulations. [MAY NEED FIGURES]
+
+After the modified Au-butanethiol surface systems are equilibrated
+under canonical ensemble, Packmol\cite{packmol} is used to pack
+organic solvent molecules in the previously vacuum part of the
+simulation cells, which guarantees that short range repulsive
+interactions do not disrupt the simulations. Two solvents are
+investigated, one which has little vibrational overlap with the
+alkanethiol and plane-like shape (toluene), and one which has similar
+vibrational frequencies and chain-like shape ({\it n}-hexane). The
+spacing filled by solvent molecules, i.e. the gap between periodically
+repeated Au-butanethiol surfaces should be carefully chosen so that it
+would not be too short to affect the liquid phase structure, nor too
+long, leading to over cooling (freezing) or heating (boiling) when a
+thermal flux is applied. In our simulations, this spacing is usually
+$35 \sim 60$\AA.
+
+The initial configurations generated by Packmol are further
+equilibrated with the $x$ and $y$ dimensions fixed, only allowing
+length scale change in $z$ dimension. This is to ensure that the
+equilibration of liquid phase does not affect the metal crystal
+structure in $x$ and $y$ dimensions. Further equilibration are run
+under NVT and then NVE ensembles.
+
+After the systems reach equilibrium, NIVS is implemented to impose a
+periodic unphysical thermal flux between the metal and the liquid
+phase. Most of our simulations are under an average temperature of
+$\sim$200K. Therefore, this flux usually comes from the metal to the
+liquid so that the liquid has a higher temperature and would not
+freeze due to excessively low temperature. This induced temperature
+gradient is stablized and the simulation cell is devided evenly into
+N slabs along the $z$-axis and the temperatures of each slab are
+recorded. When the slab width $d$ of each slab is the same, the
+derivatives of $T$ with respect to slab number $n$ can be directly
+used for $G^\prime$ calculations:
+\begin{equation}
+G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big|
+         \Big/\left(\frac{\partial T}{\partial z}\right)^2
+         = |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big|
+         \Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2
+         = |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big|
+         \Big/\left(\frac{\partial T}{\partial n}\right)^2
+\label{derivativeG2}
+\end{equation}
+
+\subsection{Force Field Parameters}
+Our simulations include various components. Therefore, force field
+parameter descriptions are needed for interactions both between the
+same type of particles and between particles of different species.
+
+The Au-Au interactions in metal lattice slab is described by the
+quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC
+potentials include zero-point quantum corrections and are
+reparametrized for accurate surface energies compared to the
+Sutton-Chen potentials\cite{Chen90}. 
+
+For both solvent molecules, straight chain {\it n}-hexane and aromatic
+toluene, United-Atom (UA) and All-Atom (AA) models are used
+respectively. The TraPPE-UA
+parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used
+for our UA solvent molecules. In these models, pseudo-atoms are
+located at the carbon centers for alkyl groups. By eliminating
+explicit hydrogen atoms, these models are simple and computationally
+efficient, while maintains good accuracy. However, the TraPPE-UA for
+alkanes is known to predict a lower boiling point than experimental
+values. Considering that after an unphysical thermal flux is applied
+to a system, the temperature of ``hot'' area in the liquid phase would be
+significantly higher than the average, to prevent over heating and
+boiling of the liquid phase, the average temperature in our
+simulations should be much lower than the liquid boiling point. [NEED MORE DISCUSSION]
+For UA-toluene model, rigid body constraints are applied, so that the
+benzene ring and the methyl-C(aromatic) bond are kept rigid. This
+would save computational time.[MORE DETAILS NEEDED]
+
+Besides the TraPPE-UA models, AA models for both organic solvents are
+included in our studies as well. For hexane, the OPLS
+all-atom\cite{OPLSAA} force field is used. [MORE DETAILS] 
+For toluene, the United Force Field developed by Rapp\'{e} {\it et
+  al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS]
+
+The capping agent in our simulations, the butanethiol molecules can
+either use UA or AA model. The TraPPE-UA force fields includes
+parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used in
+our simulations corresponding to our TraPPE-UA models for solvent. 
+ and All-Atom models [NEED CITATIONS]
+However, the model choice (UA or AA) of capping agent can be different
+from the solvent. Regardless of model choice, the force field
+parameters for interactions between capping agent and solvent can be
+derived using Lorentz-Berthelot Mixing Rule.
+
+To describe the interactions between metal Au and non-metal capping
+agent and solvent, we refer to Vlugt\cite{vlugt:cpc2007154} and derive
+other interactions which are not yet finely parametrized. [can add
+hautman and klein's paper here and more discussion; need to put
+aromatic-metal interaction approximation here]\cite{doi:10.1021/jp034405s}
+
+[TABULATED FORCE FIELD PARAMETERS NEEDED]
+
+
+[SURFACE RECONSTRUCTION PREVENTS SIMULATION TEMP TO GO HIGHER]
+
+
 \section{Results}
+[REARRANGEMENT NEEDED]
 \subsection{Toluene Solvent}
 
-The simulations follow a protocol similar to the previous gold/water
-interfacial systems. The results (Table \ref{AuThiolToluene}) show a
+The results (Table \ref{AuThiolToluene}) show a
 significant conductance enhancement compared to the gold/water
 interface without capping agent and agree with available experimental
 data. This indicates that the metal-metal potential, though not
